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For a given sequence of nonincreasing numbers, ${\bold d}=(d_1,\ldots , d_n)$, a necessary and sufficient condition is presented to characterize ${\bold d}$ when it is a degree sequence of a unique labeled simple graph. If $G$ is a graph, we consider the subgraph $G'$ of $G$ with maximum edges which is uniquely determined with respect to its degree sequence. We call the set of $E(G)\backslash E(G')$ the smallest edge defining set of $G$. This definition coincides with the similar one in design theory.
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